How do you find the integral of #x^5*e^(x^2) #?

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Answer 1 · 2018-04-25T10:50:16

#1/2x^4e^(x^2)-x^2e^(x^2)+e^(x^2)+C#

#intx^5*e^(x^2)dx#

#=1/2intx^4e^(x^2)*2xdx#

#=1/2intx^4e^(x^2)dx^2#

#x^2=u#

#d(x^2)=du#

#=1/2intx^4e^(x^2)dx^2=1/2intu^2e^udu#

#=1/2intu^2d(e^u)#

#=1/2(u^2e^u-int2ue^udu)#

#=1/2(u^2e^u-2intud(e^u))#

#=1/2(u^2e^u-2ue^u-(-2inte^udu))#

#=1/2u^2e^u-ue^u+e^u+C#

Reverse The Substitution

#=1/2x^4e^(x^2)-x^2e^(x^2)+e^(x^2)+C#